Redish & Bing, GIREP Conference Poster (2009)

Using Math in Physics: Warrants and Epistemological Frames

Edward F. Redish and Thomas J. Bing

Prepared in conjunction with Symposium, “Mathematization in Physics Lessons: Problems and Perspectives”, R. Karam and G. Pospiech, organizers. GIREP meeting, Leicester, UK, 18. August, 2009.

Abstract: Mathematics is an essential component of university level science, but it is more complex than a straightforward application of rules and calculation. Using math in science critically involves the blending of ancillary information with the math in a way that both changes the way that equations are interpreted and provides metacognitive support for recovery from errors. We have made ethnographic observations of groups of students solving physics problems in classes ranging from introductory algebra based physics to graduate quantum mechanics. These lead us to conjecture that expert problem solving in physics requires the development of the complex skill of mixing different classes of warrants – the ability to blend physical, mathematical, and computational reasons for constructing and believing a result. In order to analyze student behavior along this dimension, we have created analytical tools including epistemic frames and games. These should provide a useful lens on the development of problem solving skills and permit an instructor to recognize the development of sophisticated problem solving behavior even when the student makes mathematical errors.

(List of references)

Bing, PhD Dissertation (2008)

An Epistemic Framing Analysis of Upper-Level Physics Students' Use of Mathematics

T. J. Bing, Ph.D. Dissertation, E. F. Redish (advisor), (2008). (html TOC and abstract)

Abstract: Mathematics is central to a professional physicist's work and, by extension, to a physics student's studies. It provides a language for abstraction, definition, computation, and connection to physical reality. This power of mathematics in physics is also the source of many of the difficulties it presents students. Simply put, many different activities could all be described as "using math in physics". Expertise entails a complicated coordination of these various activities. This work examines the many different kinds of thinking that are all facets of the use of mathematics in physics. It uses an epistemological lens, one that looks at the type of explanation a student presently sees as appropriate, to analyze the mathematical thinking of upper level physics undergraduates. Sometimes a student will turn to a detailed calculation to produce or justify an answer. Other times a physical argument is explicitly connected to the mathematics at hand. Still other times quoting a definition is seen as sufficient, and so on. Local coherencies evolve in students' thought around these various types of mathematical justifications. We use the cognitive process of framing to model students' navigation of these various facets of math use in physics.

We first demonstrate several common framings observed in our students' mathematical thought and give several examples of each. Armed with this analysis tool, we then give several examples of how this framing analysis can be used to address a research question. We consider what effects, if any, a powerful symbolic calculator has on students' thinking. We also consider how to characterize growing expertise among physics students. Framing offers a lens for analysis that is a natural fit for these sample research questions. To active physics education researchers, the framing analysis presented in this dissertation can provide a useful tool for addressing other research questions. To physics teachers, we present this analysis so that it may make them more explicitly aware of the various types of reasoning, and the dynamics among them, that students employ in our physics classes. This awareness will help us better hear students' arguments and respond appropriately.

Scherr, Russ, Bing & Hodges, Phys Rev Special Topics: PER (2006)

Initiation of student-TA interactions in tutorials

R. E. Scherr, R. S. Russ, T. J. Bing & R. A. Hodges, Phys. Rev. - Special Topics: Physics Education Research 2, 020108-020116 (2006). (html link to journal article)

Abstract: At the University of Maryland we videotaped several semesters of tutorials as part of a large research project. A particular research task required us to locate examples of students calling the teaching assistants TAs over for assistance with a physics question. To our surprise, examples of this kind of interaction were difficult to find. We undertook a systematic study of TA-student interactions in tutorial: In particular, how are the interactions initiated? Do the students call the TA over for help with a particular issue, does the TA stop by spontaneously, or does the worksheet require a discussion with the TA at that point? The initiation of the interaction is of particular interest because it provides evidence of the motivation for and purpose of the interaction. This paper presents the results of that systematic investigation. We discovered that the majority of student-TA interactions in tutorial are initiated by teaching assistants, confirmed our initial observation that relatively few interactions are initiated by students, and found, further, that even fewer interactions are worksheet initiated. Perhaps most importantly, we found that our sense of who initiates tutorial interactions—based on extensive but informal observations—is not necessarily accurate. We need systematic investigations to uncover the reality of our classroom experiences.

Bing & Redish, Conference Proceedings (2008)

Using warrants as a window to epistemic framing

T. J. Bing & E. F. Redish, Proceedings of the Physics Education Research Conference, Edmonton, AB, July 2008, to be published.

Abstract: Mathematics can serve many functions in physics. It can provide a computational system, reflect a physical idea, conveniently encode a rule, and so forth. A physics student thus has many different options for using mathematics in his physics problem solving. We present a short example from the problem solving work of upper level physics students and use it to illustrate the epistemic framing process: “framing” because these students are focusing on a subset of their total math knowledge, “epistemic” because their choice of subset relates to what they see (at that particular time) as the nature of the math knowledge in play. We illustrate how looking for students’ warrants, the often unspoken reasons they think their evidence supports their mathematical claims, serves as a window to their epistemic framing. These warrants provide a powerful, concise piece of evidence of these students’ epistemic framing.

Bing & Redish, Am J Phys (2008)

Symbolic manipulators affect mathematical mindsets

T. J. Bing & E. F. Redish, Am J Phys, 76, p 418-424 (2008). (html version)

Abstract: The use of symbolic calculators such as MATHEMATICA is becoming more commonplace among upper level physics students. The presence of such powerful calculators can couple strongly to the type of mathematical reasoning students employ. These tools do not merely offer students a convenient way to perform the calculations they would have otherwise done by hand. We present examples from the work of upper level physics majors where MATHEMATICA plays an active role in focusing and sustaining their thoughts around calculation. These students still engage in powerful mathematical reasoning while they calculate, but struggle because of the narrowed breadth of their thinking. We model MATHEMATICA'S influence as an integral part of the constant feedback that occurs in how students frame, and hence focus, their work.

Bing & Redish, PER Conference Proceedings (2007)

The Cognitive Blending of Mathematics and Physics Knowledge

T. J. Bing & E. F. Redish, in Proceedings of the Physics Education Research Conference, Syracuse, NY, August 2006, AIP Conf. Proc., 883, p 26-29 (2007).

Abstract: Numbers, variables, and equations are used differently in a physics class than in a pure mathematics class. In physics, these symbols not only obey formal mathematical rules but also carry physical ideas and relations. This paper focuses on modeling how this combination of physical and mathematical knowledge is constructed. The cognitive blending framework highlights both the different ways this combination can occur and the emergence of new insights and meaning that follows such a combination. After an introduction to the blending framework itself, several examples from undergraduate physics students’ work are analyzed.